3.4.89 \(\int \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2} \, dx\) [389]

Optimal. Leaf size=330 \[ \frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

1/2*x*arccosh(a*x)^(5/2)*(-a^2*c*x^2+c)^(1/2)+5/16*arccosh(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*
x+1)^(1/2)-5/8*a*x^2*arccosh(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)-1/7*arccosh(a*x)^(7/2
)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+15/512*erf(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(
-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-15/512*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(-a
^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+15/32*x*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)

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Rubi [A]
time = 0.42, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5895, 5893, 5884, 5939, 5887, 5556, 12, 3389, 2211, 2235, 2236} \begin {gather*} \frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2),x]

[Out]

(15*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/32 + (5*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(16*a*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]) - (5*a*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*S
qrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2))/2 - (Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(7/2))/(7*a*Sqrt[-1 + a*x]*Sqrt[1
+ a*x]) + (15*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 +
a*x]) - (15*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*
x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5884

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCosh[c*x])^n/(
m + 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])
), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]

Rule 5887

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Cosh
[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(a + b*
ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps

\begin {align*} \int \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{5/2} \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \int \frac {\cosh ^{-1}(a x)^{5/2}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (5 a \sqrt {c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^{3/2} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \int \frac {\sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 a \sqrt {c-a^2 c x^2}\right ) \int \frac {x}{\sqrt {\cosh ^{-1}(a x)}} \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 \sqrt {c-a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{128 a \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {15}{32} x \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}+\frac {5 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {5 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c-a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )}{256 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}

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Mathematica [A]
time = 0.34, size = 148, normalized size = 0.45 \begin {gather*} -\frac {\sqrt {-c (-1+a x) (1+a x)} \left (-105 \sqrt {2 \pi } \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+105 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}(a x)}\right )+8 \sqrt {\cosh ^{-1}(a x)} \left (64 \cosh ^{-1}(a x)^3+140 \cosh ^{-1}(a x) \cosh \left (2 \cosh ^{-1}(a x)\right )-7 \left (15+16 \cosh ^{-1}(a x)^2\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )\right )}{3584 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2),x]

[Out]

-1/3584*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(-105*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 105*Sqrt[2*Pi]*Erf
i[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 8*Sqrt[ArcCosh[a*x]]*(64*ArcCosh[a*x]^3 + 140*ArcCosh[a*x]*Cosh[2*ArcCosh[a*x]
] - 7*(15 + 16*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \sqrt {-a^{2} c \,x^{2}+c}\, \mathrm {arccosh}\left (a x \right )^{\frac {5}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(5/2),x)

[Out]

int((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(5/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^(5/2), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**(5/2)*(-a**2*c*x**2+c)**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4368 deep

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^{5/2}\,\sqrt {c-a^2\,c\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^(5/2)*(c - a^2*c*x^2)^(1/2),x)

[Out]

int(acosh(a*x)^(5/2)*(c - a^2*c*x^2)^(1/2), x)

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